Wireless Sensor Network, 2010, 2, 148-160
doi:10.4236/wsn.2010.22020 Published Online February 2010 (http://www.SciRP.org/journal/wsn/).
A Novel Approach for Finding a Shortest Path in a Mixed
Fuzzy Network
Ali Tajdin1, Iraj Mahdavi1*, Nezam Mahdavi-Amiri2, Bahram Sadeghpour-Gildeh3,
Reza Hassanzadeh1
1
Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran
2
Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
3
Department of Mathematics and Statistics, Mazandaran University, Babolsar, Iran
E-mail: [email protected]
Received October 11, 2009; revised November 13, 2009; accepted November 14, 2009
Abstract
We present a novel approach for computing a shortest path in a mixed fuzzy network, network having various fuzzy arc lengths. First, we develop a new technique for the addition of various fuzzy numbers in a path
using α -cuts. Then, we present a dynamic programming method for finding a shortest path in the network.
For this, we apply a recently proposed distance function for comparison of fuzzy numbers. Four examples
are worked out to illustrate the applicability of the proposed approach as compared to two other methods in
the literature as well as demonstrate the novel feature offered by our algorithm to find a fuzzy shortest path
in mixed fuzzy networks with various settings for the fuzzy arc lengths.
Keywords: Fuzzy Numbers, α -Cut; Shortest Path, Dynamic Programming
1. Introduction
Determination of shortest distance and shortest path between two vertices is one of the most fundamental problems in graph theory. Let G = (V, E) be a graph with V as
the set of vertices and E as the set of edges. A path between two vertices is an alternating sequence of vertices
and edges starting and ending with vertices, and no vertices or no edges appear more than once in the sequence.
The length of a path is the sum of the weights of the
edges on the path. There may exist more than one path
between a pair of vertices. The shortest path problem is
to find the path with minimum length between a specified pair of vertices. In classical graph theory, the weight
of each edge is taken as a crisp real number. But, practically weight of each edge of the network may not be a
fixed real number and it may well be imprecise.
The shortest path problem involves addition and comparison of the edge weights. Since, the addition and
comparison between two interval numbers or between
two triangular fuzzy numbers are not alike those between
two precise real numbers, it is necessary to discuss them
at first. Interval arithmetic was developed in Moore [1].
The case of optimization with interval valued and fuzzy
constraints were discussed in Delgado et al., Ishibuchi
Copyright © 2010 SciRes.
and Tanaka, Sengupta, and Shaocheng [2–5]. Various
ranking methods for interval numbers were introduced
by several researchers. An extensive survey of the order
relations along with a new proposal are given by Sengupta and Pal [6]. There are also ranking methods for
fuzzy numbers available in the literature. Dubois and
Prade [7] introduced a ranking of fuzzy numbers in the
setting of possibility theory, and Chen [8] ranked fuzzy
numbers using maximizing and minimizing sets. Ranking of fuzzy numbers was also studied by Bortolan and
Degani, Cheng, and Delgado et al. [2,9,10].
Fuzzy graph problems were addressed in Blue et al.
and Koczy, Klein, Li et al., Lin and Chen, Okada and
Gen [11–17] paid special attention to fuzzy shortest paths.
In a recent development, Okada and Soper [18] proposed
an algorithm to find the shortest path in a network with
fuzzy edge weights. The algorithm gives a family of
non-dominated shortest paths for a specified satisfaction
level, but it does provide any guideline to the decisionmaker to choose the best amongst the paths according to
his/her view; i.e., optimistic, pessimistic, etc.
The shortest path (SP) problem has received lots of
attention from researchers in the past decades, because it
is important to many applications such as communication,
transportation, scheduling and routing. In a network, the
WSN
A. TAJDIN
arc length may represent time or cost. Conventionally, it
is assumed to be crisp. However, it is difficult for decision makers to specify the arc lengths. For example, using the same modem to transmit the data from node a to
b in a network, the data transmission time may not be the
same every time. Therefore, in real world, the arc length
could be uncertain. Fuzzy set theory, as proposed by
Zadeh [19], is frequently utilized to deal with uncertainty.
Zadeh presented the possibility theory using membership
functions to describe uncertainties.
Considering a directed network that is composed of a
finite set of nodes and a set of directed arcs, we denote
each arc by an ordered pair (i, j), where i and j are two
different nodes. The arc length represents the distance
needed to traverse (i, j) from node i to j. We denote it by
l(i, j) or L(i, j), when it is crisp or fuzzy, respectively. We
call L(i, j) as “fuzzy arc length”.
The shortest path problem is the following: given a
weighted, directed graph and two special vertices s and t,
compute the weight of the shortest path between s and t.
The shortest path problem is one of the most fundamental network optimization problems. This problem comes
up in practice and arises as a subproblem in many network optimization algorithms. Algorithms for this problem have been studied for a long time [20–22]. However,
advances in the method and theory of shortest path algorithms are still being made [23–25].
In the network we consider here, the lengths of the
arcs are assumed to represent transportation times or
costs rather than geographical distances. As time or cost
fluctuate with traffic conditions, payload and so on, it is
not practical to represent the arcs as crisp values. Thus, it
is appropriate to utilize fuzzy numbers based on fuzzy set
theory. In proposing an algorithm for solving the problem, we are first faced with the comparison or ordering
of fuzzy numbers, a task not considered to be routine.
For this reason, fuzzy shortest path problems have rarely
been studied despite their potential application to many
problems [18,26]. The problem turns out to be even more
complicated in our more general case of allowing various
fuzzy arc lengths.
Here, we propose a new approach and an algorithm to
find a shortest path in a mixed network having various
fuzzy arc lengths. The remainder of the paper is organized as follows. In Section 2, basic concepts and definitions are given. A dynamic programming algorithm for
finding a fuzzy shortest path in a network is presented in
Section 3. There, we make use of α -cuts for computing
approximations for the addition of two different types of
fuzzy numbers and apply a distance function for the
comparison of fuzzy numbers. Comparative examples
are given in Section 4. Section 5 works out an example
to show the novel feature of our algorithm to find fuzzy
shortest paths in mixed fuzzy networks with various settings for the fuzzy arc lengths. We conclude in Section 6.
Copyright © 2010 SciRes.
ET
AL.
149
2. Concepts and Definitions
We start with basic definitions of some well-known
fuzzy numbers.
Definition 1. An LR fuzzy number is represented by
~
A = (m, α , β ) LR , with the membership function, µ a~ ( x) ,
defined by the expression,
 m − x
x≤m
L 

  α 
µ a~ ( x) = 
 R  x − m  x ≥ m,
  β 
where L and R are non-increasing functions from R+ to
[0,1], L(0)=R(0)=1, m is the center, α is the left spread
and β is the right spread.
Note that if L(x)=R(x)=1-x with 0<x<1, then x is a
triangular fuzzy number and is represented by the triplet
a~ = (a1 , a 2 , a3 ) , with the membership function, µ a~ ( x) ,
defined by
x ≤ a1
 0
 x − a1
a1 < x ≤ a 2

a −a
µ a~ ( x) =  2 1
a −x
 3
a 2 < x ≤ a3
a
 3 − a2
 0
x > a3 .
A triangular fuzzy number is shown in Figure 1.
Definition 2. A trapezoidal fuzzy number a~ is
shown by a~ = (a1 , a 2 , a3 , a 4 ) , with the membership
function as follows:
 0
 x − a1

 a 2 − a1
µ a~ ( x) =  1
 a4 − x

 a 4 − a3
 0
x ≤ a1
a1 ≤ x ≤ a 2
a 2 ≤ x ≤ a3
a3 ≤ x ≤ a 4
a 4 ≤ x.
Figure 1. A triangular fuzzy number.
WSN
A. TAJDIN
150
A general trapezoidal fuzzy number is shown in Figure 2. It is apparent that a triangular fuzzy number is a
special trapezoidal fuzzy number with a 2 = a3 .
Definition 3. If L(x)=R(x)= e − x 2 , with x ∈ ℜ ,
then x is a normal fuzzy number that is shown by
(m, σ ) and the corresponding membership function is
defined to be:
µ a~ ( x) = e
−(
x −m 2
)
σ
, x∈ℜ ,
where m is the mean and σ is the standard deviation. A
normal fuzzy number is shown in Figure 3.
Definition 4. The α -cut and strong α -cut for a
~
~
~
fuzzy set A are shown by Aα and Aα+ , respectively,
and for α ∈ [0,1] are defined to be:
~
Aα = x µ A~ ( x) ≥ α , x ∈ X ,
~
Aα+
{
= {x
µ A~
}
( x) > α , x ∈ X },
where X is the universal set.
~
Upper and lower bounds for any α -cut ( Aα ) are
~
~
shown by sup Aα and inf Aα , respectively. Here, we
assume that the upper and lower bounds of α -cuts are
ET
AL.
~
finite values and for simplicity we show sup Aα by
~
Aα+ and inf Aα by Aα− .
2.1. Computing α -Cuts for Fuzzy Numbers
For an LR fuzzy number with L and R as invertible functions, the α -cuts are:
m−x
m−x
= L−1 (α ) ⇒ x = m − β L−1 (α ),
α = L[
] ⇒
β
β
x−m
x−m
= R −1 (α ) ⇒ x = m + γR −1 (α ).
α = R[
] ⇒
γ
γ
For specific L and R functions, the
discussed. Let a~ = (a1 , a 2 , a3 , a 4 )
fuzzy number. The α -cut for a~ ,
as:
 0
 x−a
1

 a2 − a1

µa% ( x) =  1
 a −x
 4
 a4 − a3

 0
α
x ≤ a1
a1 ≤ x ≤ a2
a2 ≤ x ≤ a3
a3 ≤ x ≤ a4
a4 ≤ x
α=
x − a1
⇒
a2 − a1
x = (a2 − a1 )α + a1
α=
a4 − x
⇒
a4 − a3
x = a4 − (a4 − a3 )α
⇒
a~α+ = a 4 − (a 4 − a3 )α

⇒ a~α = 
 a~ − = (a − a )α + a
2
1
1
 α
Figure 2. A trapezoidal fuzzy number.
following cases are
be a trapezoidal
(a~ ) , is computed
, 0 ≤ α ≤ 1.
(1)
Note that the α -cut for triangular fuzzy numbers is
simply obtained by using the above equations considering a 2 = a3 :
 a~ + = a3 − (a3 − a 2 )α
a~α =  ~ α−
, 0 ≤ α ≤ 1.
(2)
aα = (a 2 − a1 )α α + a1
For a~ = (m, σ ) , a normal fuzzy number, a~α , is
computed as:
−(
m−x
)
σ 2
−(
x−m
)
σ 2
α =e
α =e
Figure 3. A normal fuzzy number.
Copyright © 2010 SciRes.
⇒
⇒
m− x
σ
x −m
− ln(α) =
σ
− ln(α) =
a~αL = m − σ − ln α

⇒ a~α = 
a~ R = m + σ − ln α
 α
, 0 < α ≤ 1.
(3)
WSN
A. TAJDIN
ET
AL.
151
MV% = Min value( a% , b% )
2.2. Fuzzy Sum Operators
= (min(a1 , b1 ), min(a2 , b2 ), min(a3 , b3 ), min(a4 , b4 )).
(4)
Here, we propose an approach for summing various
fuzzy numbers approximately using α -cuts. The approximation is based on fitting an appropriate model for
the sum using α -cuts of the addition by a set of α i
values. Let us divide the α -interval [0,1] into n equal
subintervals by letting α 0 = 0 , α i = α i −1 + ∆α i ,
i=1,…,n with ∆α = 1 , i=1,…,n. This way, we have a
i
n
set of n+1 equidistant α i points. For addition of two
different fuzzy numbers, we add the set of corresponding
α i -cut points of the two numbers to yield the α i -cuts of
the sum as an approximation for the fuzzy addition.
It is evident that, for non-comparable fuzzy numbers
~
a~ and b , the fuzzy min operation results in a fuzzy
number different from both of them. For example, for
~
a~ = (5,10,13,19) and b = (6,9,15,20) , we get from (4) a
~
fuzzy MV = (5, 9,13,19) which is different from both
~
a~ and b . To alleviate this drawback, we use a
method based on the distance between fuzzy numbers.
We use the distance function introduced by Sadeghpour-Gildeh and Gien [27]. The main advantages of this
distance function are the generality of its usage on
various fuzzy numbers, and its reliability in distinguishing unequal fuzzy numbers. Indeed, the use of this
distance function worked out to be quite appropriate for
our approach here as well as in a different context
before where we considered the arc lengths in the
network to be all of the same type (see Mahdavi et al.
[28]).
The proposed D p , q -distance, indexed by parameters
3. An Algorithm for Fuzzy Shortest Path in a
Network
3.1. Distance between Fuzzy Numbers
Knowing that we can obtain a good approximation of the
addition of various fuzzy numbers by use of α -cuts, we
compute the distance between two fuzzy numbers using
~
the resulting points from the α -cuts. For a~ and b as
two different fuzzy numbers, we use a new fuzzy ranking
method for the fuzzy numbers. Let us consider the fuzzy
min operation to be defined as follows
1 < p < ∞ and 0 < q < 1 , between two fuzzy numbers
a% and b% is a nonnegative function given by:
The analytical properties of D p , q depend on the first
1

1
1
p
p


 (1 − q ) a − − b − d α + q a + − b + d α p ,
α
α
α
α


0
0

D p , q ( a% , b% ) =  

−
−
+
+
aα − bα
,
 (1 − q ) sup aα − bα + q 0inf
<α ≤1
0 <α ≤1

∫
(
parameter p, while the second parameter q of
D characterizes the subjective weight attributed to
p,q
the end points of the support; i.e., ( aα+ , aα− ) of the fuzzy
numbers. If there is no reason for distinguishing any
side of the fuzzy numbers, then D 1 is recommended.
p,
)
∫
(
(5)
p = ∞.
n
~ 
D p , q (a~, b ) = (1 − q )
aα−i − bα−i
i =1

∑
p
n
+q
∑a
+
αi
− bα+i
1
p p
 .

(6)
1
If q = , p = 2 , then the above equation turns into:
2
2
Having q close to 1 results in considering the right side
of the support of the fuzzy numbers more favorably.
Since the significance of the end points of the support
of the fuzzy numbers is assumed to be the same, then
we consider q = 1 .
)
p < ∞,
D
~ ~ =
1 (a , b )
2,
2
1
2
n
∑
i =1
(aα−i − bα−i ) 2 +
i =1
1
2
n
∑ (a
i =1
+
αi
− bα+i ) 2 ,
~ and b~ with correspondFor two fuzzy numbers a
ing α i -cuts, the D p ,q distance is approximately pro-
(7)
~
~
To compare two fuzzy arc lengths a and b with
α i -cuts as their approximations, since they are supposed
to represent positive values, we compare them with
~
MV = (0, 0, ....,0) . In fact, we use (7) to compute
~
D 1 (a~,0) and D 1 (b ,0) and use these values for
portional to:
comparison of the two numbers.
2
Copyright © 2010 SciRes.
2,
2
2,
2
WSN
A. TAJDIN
152
3.2. An Algorithm for Computing s Shortest
Fuzzy Path
The following dynamic programming algorithm is for
computing the shortest path in a network. The algorithm
is based on Floyd’s dynamic programming method to
find a shortest path, if it exists, between every pair of
nodes i and j in the network (see Floyd [29]).
We make use of the following optimal value function
f k (i, j ) and the corresponding labeling function Pk (i, j ) :
 Pk −1 (i, j )
Pk (i, j ) = 
 Pk −1 (k , j )
node i to node j then let Pk (i, j ) = i .
Compute
the
value
of
[ f k −1 (i, j ), f k −1 (i, k ) + f k −1 (k , j )]
f k (i, j ) = length of the shortest path from node i to
node j when the path is considered to use only the nodes
from the set of nodes {1, ...., k } .
Pk (i, j ) = the last intermediate node on the shortest
path from node i to node j using {1, ...., k } as intermediate
node.
The dynamic updating for the optimal path length and
its corresponding labeling are:
f k (i, j ) = min{ f k −1 (i, j ) , f k −1 (i, k ) + f k −1 (k , j )} ,
otherwise.
the set of arcs.
~
~
Step 1: Let k=0 and f k (i, j ) = d ij , for all (i, j ) ∈ A ,
~
f k = (i, j ) = ∞ , for all (i, j ) ∉ A . If an arc exists from
2.1
AL.
if k is not on shortest path from i to j using {1,...,k}
We are now ready to give the steps of the algorithm.
Algorithm: A dynamic programming method for
computing a shortest path in a fuzzy network G = (V , A) ,
where V is the set of nodes with V = N , and A is
Step 2: Let k = k + 1 .
Do
the
following
i = 1, 2, 3, ...., N , j = 1, 2, 3, ...., N , i ≠ j.
ET
steps
for
f k (i, j ) = min
(for the addition,
our proposed method discussed in Subection 2.2 and for
comparison of fuzzy numbers the D p , q method of
Subection 3.1 are applied).
2.2 If node k is not on the shortest path using nodes
{1, 2, ..., k} as intermediate nodes, then let Pk (i, j ) =
resulting in 2n(N)(N-1)2 additions. For comparisons, we
have (2n+1)N(N-1)2 additions and (2n+1) N(N-1)2 multiplications using (7). Therefore, the total needed operations are (6n+2) N(N-1)2 additions and multiplications,
with N(N-1)2 comparison
4. Comparative Examples
Here, we illustrate examples for a comparison of our
proposed method and two other approaches.
Example 1:
Consider the following network Figure 4 considered by
Chuang and Kung [30]. The triangular arc lengths are
presented in Table 1. The results obtained by the ap~
proach ( f and P6 ) of Chuang and Kung [30] are
6
shown in Tables 2 and 3.
The shortest path and the corresponding length using the
proposed approach in Chuang and Kung [30] are reported below:
Shortest path from 1 to 6 : 1 → 2 → 4 → 6 .
Shortest path length from 1 to 6 : (177,195, 256)
Pk −1 (i, j ) else let Pk (i, j ) = Pk −1 (k , j )
Step 3: If k < N then go to Step 2.
Step 4: Obtain the shortest path using the Pk (i, j ) . If
f N (i, j ) = ∞ , then there is no path between i and j. The
shortest path from node i to j, if it exists, is identified
backwards and read by the nodes: j, PN (i, j ) = k followed by PN (i, k ), ...., PN (i, l ) = i , where l is the node
immediately after i in the path.
3.3. Termination and Complexity of the
Algorithm
The proposed algorithm terminates after N outer iterations corresponding to k. A total of N(N-1)2 additions and
comparisons are needed for every k. For each addition, n
fuzzy additions for the α i -cuts should be performed
Copyright © 2010 SciRes.
Figure 4. The network for Example 1.
Table 1. The arc lengths for example 1.
Arc
lengths
Arc
lengths
Arc
lengths
(1,2)
(33,45,50)
(1,3)
(42,57,61)
(2,3)
(50,52,61)
(2,4)
(56,58,72)
(2,5)
(51,79,85)
(3,5)
(43,55,60)
(4,5)
(32,40,46)
(4,6)
(88,92,134)
(5,6)
(75,110,114)
WSN
A. TAJDIN
ET
AL.
153
Table 2. The ~f 6 values obtained by the approach of Chuang and Kung.
i/j
1
2
3
4
5
6
1
-
(33,45,50)
(42,57,61)
(89,103,122)
(85,112,121)
(177,195,256)
2
-
-
(50,52,61)
(56,58,72)
(51,79,85)
(144,150,206)
3
-
-
-
-
(43,55,60)
(118,165,174)
4
-
-
-
-
(32,40,46)
(88,92,134)
5
-
-
-
-
-
(75,110,114)
6
-
-
-
-
-
-
Table 3. The P6 values obtained by the approach of
Chuang and Kung.
i/j
1
2
3
4
5
6
1
-
1
1
2
3
4
2
-
-
2
2
2
4
3
-
-
-
-
3
5
4
5
6
-
-
-
-
4
-
4
5
-
Here, we solve the same problem using our proposed
algorithm givenin Subection 3.2 using the ranking
method of Sadeghpour-Gildeh and Gien [27]. The results
~
of the proposed approach for f 6 and P6 are given in
Tables 4 and 5.
Here, the shortest path obtained and the corresponding
length are exactly the same as the ones we obtained by
the approach of Chuang and Kung [30].
Example 2:
Consider the following network Figure 5 considered
~
Table 4. The f 6 values obtained by our proposed algorithm
i/j
1
2
3
4
5
6
1
-
(33,45,50)
(42,57,61)
(89,103,122)
(85,112,121)
(177,195,256)
2
-
-
(50,52,61)
(56,58,72)
(51,79,85)
(144,150,206)
3
-
-
-
-
(43,55,60)
(118,165,174)
4
-
-
-
-
(32,40,46)
(88,92,134)
5
-
-
-
-
-
(75,110,114)
6
-
-
-
-
-
-
by Hernandes et al. [32]. The fuzzy triangular arc lengths
~
are given in Table 6. The results ( f 6 and P6 ) for the
approach of Hernandes et al. [32] are given in Tables 7
and 8.
The shortest path and the corresponding length using
the proposed approach of Hernandes et al. [32] are reported below:
Shortest path from 1 to 11 : 1 → 9 → 7 → 11.
Shortest path length from 1 to 11 : (860, 902, 990 )
Table 5. The P6 values obtained by our proposed algorithm.
1
-
i/j
1
2
3
4
5
6
Clearly, the proposed algorithm computes almost the
same solution as obtained by Hernandes et al. [32].
Copyright © 2010 SciRes.
4
2
2
-
5
3
2
3
4
-
6
4
4
5
4
5
-
11
7
6
9
~
proposed approach ( f11 and P11 ) are given in Tables 9
Shortest path length from 1 to 11 : (840, 882, 990 ) .
3
1
2
-
10
We solved the same problem using our proposed algorithm of Subsection 3.2 using the ranking method of
Sadeghpour-Gildeh and Gien [27]. The results of our
and 10.
The shortest path and the corresponding length are reported below:
Shortest path from 1 to 11 : 1 → 9 → 7 → 11 .
2
1
-
8
4
1
3
5
2
Figure 5. The network for Example 2.
WSN
A. TAJDIN
154
ET
AL.
Table 6. The arc lengths for Example 2.
Arc
lengths
Arc
lengths
Arc
lengths
(1,2)
(800,820,840)
(3,5)
(730,748,770)
(8,4)
(710,730,735)
(1,3)
(350,361,370)
(3,8)
(425,443,465)
(8,7)
(230,242,255)
(1,6)
(650,677,683)
(4,5)
(190,199,210)
(9,7)
(120,130,150)
(1,9)
(290,300,350)
(4,6)
(310,340,360)
(9,8)
(130,137,145)
(1,10)
(420,450,470)
(4,11)
(710,740,770)
(9,10)
(230,242,260)
(2,3)
(180,186,193)
(5,6)
(610,660,690)
(10,7)
(330,342,350)
(2,5)
(495,510,525)
(6,11)
(230,242,260)
(10,11)
(1250,1310,1440)
(2,9)
(900,930,960)
(7,6)
(390,410,440)
(3,4)
(650,667,883)
(7,11)
(450,472,490)
~
Table 7. The f11(i, j ) values obtained by Hernandes et al.
i/j
1
2
3
4
5
6
7
1
-
(800,820,840)
(350,361,370)
(1000,1028,1253)
(1080,1109,1140)
(650,677,683)
(410,430,500)
2
-
-
(180,186,193)
(830,853,1076)
(495,510,525)
(1105,1170,1215)
(835,871,913)
3
-
-
-
(650,667,883)
(730,748,770)
(960,1007,1243)
(655,685,720)
4
-
-
-
-
(190,199,210)
(310,340,360)
-
5
-
-
-
-
-
(610,660,690)
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
(390,410,440)
-
8
-
-
-
(710,730,735)
(900,929,945)
(620,652,695)
(230,242,255)
9
-
-
-
(840,867,880)
(1030,1066,1090)
(510,540,590)
(120,130,150)
10
-
-
-
-
-
(720,752,790)
(330,342,350)
11
-
-
-
-
-
-
-
Table 7. continued.
i/j
8
9
10
11
1
(420,437,495)
(290,300,350)
(420,450,470)
(860,902,990)
2
(605,629,658)
(900,930,960)
(1130,1172,1220)
(1285,1343,1403)
3
(425,443,465)
-
-
(1105,1157,1210)
4
-
-
-
(540,582,620)
5
-
-
-
(840,902,950)
6
-
-
-
(230,242,260)
7
-
-
-
(450,472,490)
8
-
-
-
(680,714,745)
9
(130,137,145)
-
(230,242,260)
(570,602,640)
10
-
-
-
(780,814,840)
11
-
-
-
-
Example 3: A wireless sensor network
Consider a mobile service company which handles 23
geographical centers. A configuration of a telecommunication network is presented in Figure 6. Assume that the
distance between any two centers is a trapezoidal fuzzy
number (the arc lengths are given in Table 11). The
company wants to find a shortest path for an effective
Copyright © 2010 SciRes.
message flow amongst the centers.
~
The results obtained by our approach ( f 23 and P23 )
are given in Tables 12 and 13.
The shortest path and the corresponding length are reported below:
Shortest path from 1 to 23 : 1 → 5 → 11 → 17 → 21 → 23 .
WSN
A. TAJDIN
ET
Shortest path length from1 to 23 : (38, 49, 58, 65) .
AL.
155
mixed fuzzy network with 4 nodes as shown in Figure 7
is considered, where the arc lengths are considered to be
a mixture of trapezoidal and normal fuzzy numbers.
Example 4: Consider the mixed fuzzy network in Figure 4 with four nodes and five arcs having two trapezoidal
and three normal arc lengths as specified in Table 14.
~
Step 1: We gain the ~
f k (i, j ) = d ij for k=0 as speci-
5. Discussion
We can also apply the proposed algorithm to networks
having possibly a mixture of different fuzzy numbers as
arc lengths. To see how the steps of the proposed algorithm are carried out on such networks, a small sized
fied in Table 14.
Table 8. The P11(i, j ) values obtained by Hernandes et al.
1
-
i/j
1
2
3
4
5
6
7
8
9
10
11
2
1
-
3
1
2
-
4
3
3
3
8
8
-
5
3
2
3
4
4
8
-
6
1
5
4
4
5
7
7
7
7
-
7
9
8
8
8
9
10
-
8
9
3
3
9
-
9
1
2
-
10
1
9
9
-
11
9
8
8
6
6
6
7
7
7
7
-
~
Table 9. The f11(i, j ) values obtained by our proposed algorithm.
i/j
1
2
3
4
5
6
7
8
9
10
11
1
-
2
(800,820,840)
-
3
(350,361,370)
(180,186,193)
-
4
(1000,1028,1253)
(830,853,1076)
(650,667,883)
(710,730,735)
(840,867,880)
-
5
(1080,1109,1140)
(495,510,525)
(730,748,770)
(190,199,210)
(900,929,945)
(1030,1066,1090)
-
6
(650,677,683)
(1105,1170,1215)
(960,1007,1243)
(310,340,360)
(610,660,690)
(390,410,440)
(620,652,695)
(510,540,590)
(720,752,790)
-
7
(410,430,500)
(835,871,913)
(655,685,720)
(230,242,255)
(120,130,150)
(330,342,350)
-
Table 9. continued.
i/j
1
2
3
4
5
6
7
8
9
10
11
8
(420,437,495)
(605,629,658)
(425,443,465)
(130,137,145)
-
9
(290,300,350)
(900,930,960)
-
10
(420,450,470)
(1130,1172,1220)
(230,242,260)
-
11
(840,882,990)
(1265,1323,1403)
(1085,1137,1210)
(540,582,620)
(840,902,950)
(230,242,260)
(430,452,490)
(660,694,745)
(550,582,640)
(760,794,840)
-
Table 10. The P11(i, j ) values obtained by our proposed algorithm.
i/j
1
2
3
4
5
6
7
8
9
10
11
Copyright © 2010 SciRes.
1
-
2
1
-
3
1
2
-
4
3
3
3
8
8
-
5
3
2
3
4
4
8
-
6
1
5
4
4
5
7
7
7
7
-
7
9
8
8
8
9
10
-
8
9
3
3
9
-
9
1
2
-
10
1
9
9
-
11
9
8
8
6
6
6
7
7
7
7
-
WSN
A. TAJDIN
156
ET
AL.
Table 11. The arc lengths for Example 3.
Arc
lengths
Arc
lengths
Arc
lengths
(1,2)
(1,5)
(3,8)
(5,8)
(6,9)
(7,11)
(9,16)
(11,14)
(12,15)
(14,21)
(16,20)
(18,21)
(19,22)
(22,23)
(12,13,15,17)
(7,8,9,10)
(10,11,16,17)
(6,9,11,13)
(6,8,10,11)
(6,7,8,9)
(6,7,9,10)
(8,9,11,13)
(12,14,15,16)
(11,12,13,14)
(9,12,14,16)
(15,17,18,19)
(15,16,17,19)
(4,5,6,8)
(1,3)
(2,6)
(4,7)
(5,11)
(6,10)
(8,12)
(10,16)
(11,17)
(13,15)
(15,18)
(17,20)
(18,22)
(20,23)
(9,11,13,15)
(5,10,15,16)
(17,20,22,24)
(7,10,13,14)
(10,11,14,15)
(5,8,9,10)
(12,13,16,17)
(6,9,11,13)
(10,12,14,15)
(8,9,11,13)
(7,10,11,12)
(3,5,7,9)
(13,14,16,17)
(1,4)
(2,7)
(4,11)
(5,12)
(7,10)
(8,13)
(10,17)
(12,14)
(13,19)
(15,19)
(17,21)
(18,23)
(21,23)
(8,10,12,13)
(6,11,11,13)
(6,10,13,14)
(10,13,15,17)
(9,10,12,13)
(3,5,8,10)
(15,19,20,21)
(13,14,16,18)
(17,18,19,20)
(5,7,10,12)
(6,7,8,10)
(5,7,9,11)
(12,15,17,18)
~
Table 12. The f 23 (i, j ) values obtained by our algorithm.
i/j
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
1
-
2
(12,13,15,17)
-
3
(9,11,13,15)
-
4
(8,10,12,13)
-
5
(7,8,9,10)
-
6
(17,23,30,33)
(5,10,15,16)
-
7
(18,24,26,30)
(6,11,11,13)
(17,20,22,24)
-
8
(13,17,20,23)
(10,11,16,17)
(6,9,11,13)
-
Table 12. continued.
i/j
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
9
(23,31,40,44)
(11,18,25,27)
(6,8,10,11)
-
Copyright © 2010 SciRes.
10
(27,34,38,43)
(15,21,23,26)
(26,30,34,37)
(10,11,14,15)
(9,10,12,13)
-
11
(14,18,22,24)
(12,18,19,22)
(6,10,13,14)
(7,10,13,14)
(6,7,8,9)
-
12
(17,21,24,27)
(15,19,25,27)
(10,13,15,17)
(5,8,9,10)
-
13
(16,22,28,33)
(13,16,24,27)
(9,14,19,23)
(3,5,8,10)
-
14
(22,27,33,37)
(20,27,30,35)
(28,33,41,45)
(14,19,24,27)
(15,19,24,27)
(14,16,19,22)
(18,22,25,28)
(8,9,11,13)
(13,14,16,18)
-
15
(29,35,39,43)
(23,28,38,42)
(22,27,30,33)
(13,17,22,25)
(12,14,15,16)
(10,12,14,15)
-
WSN
A. TAJDIN
ET
AL.
157
Table 12. continued.
i/j
16
17
18
19
20
21
22
23
1
(29,38,49,54)
(20,27,33,37)
(37,44,50,56)
(33,40,47,53)
(27,37,44,49)
(26,34,41,47)
(40,49,57,65)
(38,49,58,65)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
(17,25,34,37)
(38,43,50,54)
(12,15,19,21)
(21,23,28,30)
(6,7,9,10)
(12,13,16,17)
-
(18,27,30,35)
(12,19,24,27)
(13,19,24,27)
(25,30,34,36)
(12,16,19,22)
(15,19,20,21)
(6,9,11,13)
-
(31,37,49,55)
(30,36,41,46)
(21,26,33,38)
(20,23,26,29)
(18,21,25,28)
(8,9,11,13)
-
(30,34,43,47)
(26,32,38,43)
(20,23,27,30)
(17,21,25,28)
(17,18,19,20)
(5,7,10,12)
-
(25,37,41,47)
(19,29,35,39)
(20,29,35,39)
(21,27,33,37)
(19,26,30,34)
(15,19,23,26)
(21,25,30,33)
(13,19,22,25)
(9,12,14,16)
(7,10,11,12)
-
(24,34,38,45)
(39,45,54,59)
(18,26,32,37)
(19,26,32,37)
(31,37,42,46)
(18,23,27,32)
(29,34,38,42)
(21,26,28,31)
(12,16,19,23)
(24,26,29,32)
(33,38,43,47)
(11,12,13,14)
(23,26,29,32)
(6,7,8,10)
(15,17,18,19)
-
(34,42,56,64)
(33,41,48,55)
(24,31,40,47)
(23,28,33,38)
(21,26,32,37)
(11,14,18,22)
(3,5,7,9)
(15,16,17,19)
-
(36,49,55,63)
(36,44,58,66)
(30,41,49,55)
(31,41,49,55)
(34,41,49,54)
(30,38,44,50)
(26,33,42,49)
(28,33,39,43)
(34,39,46,50)
(24,31,36,41)
(25,30,35,40)
(23,28,34,39)
(23,27,30,32)
(13,16,20,24)
(22,26,30,33)
(18,22,25,28)
(5,7,9,11)
(19,21,23,27)
(13,14,16,17)
(12,15,17,18)
(4,5,6,8)
23
-
-
-
-
-
-
-
-
Table 13. The P23 (i, j ) values obtained by our algorithm.
i/j
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
1
-
1
1
1
1
2
2
5
6
7
5
5
8
11
12
9
11
15
13
17
17
18
21
2
-
-
-
-
-
2
2
-
6
7
7
-
-
11
-
9
11
-
-
17
17
-
21
3
-
-
-
-
-
-
-
3
-
-
-
8
8
12
13
-
-
15
13
-
14
18
18
4
-
-
-
-
-
-
4
-
-
7
4
-
-
11
-
10
11
-
-
17
17
-
21
5
-
-
-
-
-
-
-
5
-
-
5
5
8
11
12
-
11
15
13
17
17
18
21
6
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
9
10
-
-
16
17
-
20
7
-
-
-
-
-
-
-
-
-
7
7
-
-
11
-
10
11
-
-
17
17
-
21
8
-
-
-
-
-
-
-
-
-
-
-
8
8
12
13
-
-
15
13
-
14
18
18
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
16
-
-
20
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
10
10
-
-
16
17
-
20
11
-
-
-
-
-
-
-
-
-
-
-
-
-
11
-
-
11
-
-
17
17
-
21
12
-
-
-
-
-
-
-
-
-
-
-
-
-
12
12
-
-
15
15
-
14
18
18
13
-
-
-
-
-
-
-
-
-
-
-
-
-
-
13
-
-
15
13
-
18
18
18
14
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
14
-
21
15
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
15
15
-
18
18
18
16
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
16
-
-
20
17
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
17
17
-
21
18
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
18
18
18
19
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
19
22
20
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
20
21
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
21
22
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
22
23
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Copyright © 2010 SciRes.
WSN
A. TAJDIN
158
6
9
16
20
10
2
7
17
1
4
21
23
11
14
18
3
12
5
13
19
Figure 6. The telecommunication network proposed for
Example 3.
Figure 7. A small sized network having various fuzzy arc
lengths.
~
f 0 (i, j ) matrix for k=0.
Table 14. The
i/j
1
2
3
4
1
-
2
(2,3,4,5)
-
3
(4,8,12,16)
(4,1)
-
AL.
If node k is not on the shortest path using {1, 2, ..., k } as
intermediate nodes, then we consider Pk (i, j ) = Pk −1 (i, j ) ,
otherwise we let Pk (i, j ) = Pk −1 (i, k ) . We now report the
results obtained for other values of k in Tables 18-23. Note
that, the sets Vi and Wi are the points obtained by α -cut
additions, where the V and W values are obtained by the
α i -cuts considering n=10. It includes 10 points for the
aα− and 10 points for the aα+ :
i
15
22
8
ET
4
2
(15,4)
(5,1)
-
Therefore, with Pk (i, j ) = i , we have Table 15.
Step 2: Here, we consider k=1 and compute the value
of f k (i, j ) = min [ f k −1 (i, j ), f k −1 (i, k ) + f k −1 (k , j )] . The
result is shown in Table 16.
Therefore, for Pk (i, j ) = i , we have Table 17.
i
V1={(4.58257,10.4174), (4.93136,10.0686),
(5.20274,9.79726), (5.44277,9.55723),
(5.66745,9.33255), (5.88528,9.11472),
(6.10278,8.89722), (6.32762,8.67238),
(6.57541,8.42459), (7,8)}
W1={(11.0303,25.9697), (12.1255,24.8745),
(12.911,24.089), (13.5711,23.4289),
(14.1698,22.8302), (14.7411,22.2589),
(15.3111,21.6889), (15.9105,21.0895),
(16.6016,20.3984), (18,19)}
V2={(4.58257,10.4174),(4.93136,10.0686),
(5.20274,9.79726), (5.44277,9.55723),
(5.66745,9.33255), (5.88528,9.11472),
(6.10278,8.89722), (6.32762,8.67238),
(6.57541,8.42459), (7,8)}
W2={(8.06515,16.9349), (8.66273,16.3373),
(9.10549,15.8945), (9.48554,15.5145),
(9.83489,15.1651), (10.1706,14.8294),
(10.5056,14.4944), (10.8552,14.1448),
(11.2508,13.7492), (12,13)}
~
Table 18. The f 2 (i, j ) matrix for k= 2.
1
-
i/j
1
2
3
4
1
-
2
1
-
3
1
2
-
1
-
2
(2,3,4,5)
-
3
(4,8,12,16)
(4,1)
-
1
-
Copyright © 2010 SciRes.
2
1
-
3
1
2
-
4
2
3
-
4
W1
(15,4)
(5,1)
-
2
1
-
3
2
2
-
4
2
2
3
-
~
Table 20. The f3 (i, j ) matrix for k= 3
4
(15,4)
(5,1)
-
Table 17. The P1(i, j ) matrix for k=1.
i/j
1
2
3
4
1
-
i/j
1
2
3
4
4
2
3
-
~
Table 16. The f1 (i, j ) matrix for k=1.
i/j
1
2
3
4
3
V1
(4,1)
-
Table 19. The P2 (i, j ) matrix for k= 2.
Table 15. The P0 (i, j ) matrix for k=0.
i/j
1
2
3
4
2
(2,3,4,5)
-
i/j
1
2
3
4
1
2
3
-
(2,3,4,5)
-
V2
(4,1)
-
W2
(9,2)
(5,1)
4
-
-
-
-
Table 21. The P3 (i, j ) matrix for k= 3.
i/j
1
2
3
4
1
-
2
1
-
3
2
2
-
4
3
3
3
-
WSN
A. TAJDIN
ET
~
Table 22. The f 4 (i, j ) matrix for k= 4.
i/j
1
2
3
4
1
-
2
(2,3,4,5)
-
3
V3
(4,1)
-
4
W3
(9,2)
(5,1)
-
1
2
3
4
1
-
1
2
3
2
-
-
2
3
3
-
-
-
3
4
-
-
-
-
V3={(4.58257,10.4174), (4.93136,10.0686),
(5.20274,9.79726), (5.44277,9.55723),
(5.66745,9.33255), (5.88528,9.11472),
(6.10278,8.89722), (6.32762,8.67238),
(6.57541,8.42459), (7,8)}
W3={(8.06515,16.9349), (8.66273,16.3373),
(9.10549,15.8945), (9.48554,15.5145),
(9.83489,15.1651), (10.1706,14.8294),
(10.5056,14.4944), (10.8552,14.1448),
(11.2508,13.7492), (12,13)}
Finally, when k = N , we identify the shortest path as
follows:
Shortest path from 1 to 4: 1-2-3-4.
Shortest path length from 1 to 4:
(8.06515,16.9349),(8.66273,16.3373),(9.10549,15.894
5),(9.48554,15.5145),(9.83489,15.1651),(10.1706,14.829
4),(10.5056,14.4944),(10.8552,14.1448),(11.2508,13.749
2),(12,13)
6. Conclusions
We presented a novel approach for computing a shortest
path in a mixed network having various fuzzy arc lengths.
First, we developed a new technique for the addition of
various fuzzy numbers in a path using α -cuts. Then,
we applied a dynamic programming method for finding a
shortest path in the network, using a recently proposed
distance function to compare fuzzy numbers in the proposed algorithm. Four comparative examples were
worked out to illustrate the applicability of our proposed
approach as compared to two other methods in the literature as well as demonstrate the additional novel feature offered by our algorithm to find a fuzzy shortest
path in mixed fuzzy networks having various settings for
the fuzzy arc lengths.
7. References
[1]
R. E. Moore, “Method and application of interval analysis,” SIAM, Philadelphia, 1997. M. Delgado, J. L. Ver-
Copyright © 2010 SciRes.
159
degay, and M. A. Vila, “A procedure for ranking fuzzy
numbers using fuzzy relations,” Fuzzy Sets and Systems,
Vol. 26, pp. 49–62, 1988.
[2]
M. Delgado, J. L. Verdegay, and M. A. Vila, “A procedure for ranking fuzzy numbers using fuzzy relations,”
Fuzzy Sets and Systems, Vol. 26, pp. 49–62, 1988.
[3]
H. Ishibuchi and H. Tanaka, “Multi-objective programming in optimization of the interval objective function,”
European Journal of Operational Research, Vol. 48, pp.
219–225, 1990.
[4]
J. K. Sengupta, “Optimal decision under uncertainty,”
Springer, New York, 1981.
[5]
T. Shaocheng, “Interval number and fuzzy number linear
programming,” Fuzzy Sets and Systems, Vol. 66, pp.
301–306. 1994.
[6]
A. Sengupta, T. K. Pal, “Theory and methodology on
comparing interval numbers,” European Journal of Operational Research, Vol. 127, 28–43, 2000.
[7]
D. Dubois and H. Prade, “Ranking fuzy numbers in the
setting of possibility theory,” Information Sciences, Vol.
30, pp. 183–224, 1983.
[8]
S. H. Chen, “Ranking fuzzy numbers with maximizing set
and minimizing set,” Fuzzy Sets and Systems, Vol. 17, pp.
113–129, 1985.
[9]
G. Bortolan and R. Degani, “A review of some methods
for ranking fuzzy subsets,” Fuzzy Sets and Systems, Vol.
15, pp. 1–19, 1985.
Table 23. The P4 (i, j ) matrix for k= 4.
i/j
AL.
[10] C.-H. Cheng, “A new approach for ranking fuzzy numbers by distance method,” Fuzzy Sets and Systems, Vol.
95, pp. 307–317, 1998.
[11] M. Blue, B. Bush, and J. Puckett, “Unified approach to
fuzzy graph problems,” Fuzzy Sets and Systems, Vol. 125,
pp. 355–368, 2002.
[12] L. T. Koczy, “Fuzzy graphs in the evaluation and optimization of networks,” Fuzzy Sets and Systems, Vol. 46, pp.
307–319, 1992.
[13] C. M. Klein, “Fuzzy Shortest Paths,” Fuzzy Sets and
Systems, Vol. 39, pp. 27–41, 1991.
[14] Y. Li, M. Gen, and K. Ida, “Solving fuzzy shortest path
problem by neural networks,” Computers Industrial Engineering, Vol. 31, pp. 861–865, 1996.
[15] K. Lin and M. Chen, “The fuzzy shortest path problem
and its most vital arcs,” Fuzzy Sets and Systems, Vol. 58,
pp. 343–353, 1994.
[16] S. Okada and M. Gen, “Order relation between intervals
and its application to shortest path problem,” Computers
& Industrial Engineering, Vol. 25, 147–150, 1993.
[17] S. Okada and M. Gen, “Fuzzy shortest path problem,”
Computers & Industrial Engineering, Vol. 27, pp. 465–
468, 1994.
[18] S. Okada and T. Soper, “A shortest path problem on a
network with fuzzy arc lengths,” Fuzzy Sets and Systems,
Vol. 109, pp. 129–140, 2000.
[19] L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, Vol. 1, pp. 3–28, 1978.
[20] R. K. Ahuja, K. Mehlhorn, J. B. Orlin, and R. E. Tarjan,
WSN
A. TAJDIN
160
“Faster algorithms for the shortest path problem,” Technical Report CS-TR-154-88, Department of Computer
Science, Princeton University, 1988.
[21] A. Andersson, T. Hagerup, S. Nilsson, and R. Raman,
“Sorting in linear time?” In Proceedings of 27th ACM
Symposium on Theory of Computing, pp. 427–436, 1995.
[22] A. V. Goldberg, “Scaling algorithms for the shortest paths
problem,” In: Proceedings of the 4th ACM-SIAM Symposium on Discrete Algorithms, pp. 222–231, 1993.
[23] Y. Asano and H. Imai, “Practical efficiency of the linear-time algorithm for the single source shortest path
problem,” Journal of the Operations Research Society of
Japan, Vol. 43, pp. 431–447, 2000.
[24] Y. Han, “Improved algorithm for all pairs shortest paths,”
Information Processing Letters, Vol. 91, pp. 245–250, 2004.
[25] S. Saunders, T. Takaoka, “Improved shortest path algorithms for nearly acyclic graphs,” Electronic Notes in
Theoretical Computer Science, Vol. 42, pp. 1–17, 2001.
[26] S. Chanas, “Fuzzy optimization in networks,” In: J.
Kacprzyk, S. A. Orlovski (Eds.), Optimization Models
Using Fuzzy Sets and Possibility Theory, D. Reidel,
Dordrecht, pp. 303–327, 1987.
Copyright © 2010 SciRes.
ET
AL.
[27] B. Sadeghpour-Gildeh, D. Gien, Q. La Distance-Dp, et al.
“Cofficient de Corrélation entre deux Variables Aléatoires floues,” Actes de LFA’01, pp. 97–102, MonseBelgium, 2001.
[28] I. Mahdavi, R. Nourifar, A. Heidarzade, and N. MahdaviAmiri, “A dynamic programming approach for finding
shortest chains in a fuzzy network,” Applied Soft Computing, Vol. 9, No. 2, pp. 503–511, 2009.
[29] R. W. Floyd, Algorithm 97, Shortest path, CACM 5, pp.
345, 1962.
[30] T.-N. Chuang, and J.-Y. Kung, The fuzzy shortest path
length and the corresponding shortest path in a network,
Computers & Operations Research, Vol. 32, 1409–1428,
2005.
[31] M. L. Fredman and R. E. Tarjan, “Fibonacci heaps and
their uses in improved network optimization algorithm,” J.
ACM 34, Vol. 3, pp. 596–615, 1987.
[32] F. Hernandes, M. T. Lamata, J. L. Verdegay, and A. Yamakami, “The shortest path problem on networks with
fuzzy parameters,” Fuzzy Sets and Systems, Vol. 158, pp.
1561–1570, 2007.
WSN